The N(H2)/I(CO) Conversion Factor: A Treatment that Includes Radiative Transfer

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THE N(H

2

)/I(CO) CONVERSION FACTOR: A TREATMENT THAT

INCLUDES RADIATIVE TRANSFER

W. F. Wall

Instituto Nacional de Astrof´ısica, Óptica y Electrónica Tonantzintla, Puebla, México

Received 2005 December 1; accepted 2006 February 7

RESUMEN

Se presenta un tratamiento que explica mejor el factor de conversión N(H2)/I(CO) y que incluye la transferencia radiativa. A primera vista, incluir la transferencia radiativa parece superfluo para una l´ınea ópticamente gruesa como CO J = 1→0. No obstante, dado que el medio interestelar es inhomogéneo, los fragmentos de gas (es decir, grumos) todav´ıa pueden ser ópticamente delgadas ha-cia sus bordes y en las alas de los pérfiles de la l´ınea. El tratamiento estad´ıstico de Martin et al. (1984) de la transferencia radiativa a través una nube molecular con grumos se usa para derivar una expresión para el factor de conversión que su-pera los defectos de las explicaciones más tradicionales basadas en Dickman et al. (1986). Por un lado, el tratamiento presentado aqu´ı posiblemente representa un avance importante al entender el factor de conversión N(H2)/I(CO) pero, por otro lado, tiene sus propios defectos, que son discutidos aqu´ı brevemente.

ABSTRACT

A treatment that better explains the N(H2)/I(CO) conversion factor is given that includes radiative transfer. At first glance, involving radiative transfer seems superfluous for an optically thick line such as CO J = 1→0 line. However, given that the interstellar medium is inhomogeneous, the individual gas fragments (i.e., clumps) can still be optically thin toward their edges and in the wings of their line profiles. The statistical treatment by Martin et al.(1984) of the radiative transfer through a clumpy molecular cloud is used to derive an expression for the conversion factor that overcomes the shortcomings in the more traditional explanations based on Dickman et al. (1986). While possibly representing important step forward in understanding the N(H2)/I(CO) conversion factor, the treatment here also has shortcomings of its own that are briefly discussed.

Key Words:ISM: MOLECULES — ISM: DUST — ISM: ORION

1. INTRODUCTION

In studies of the interstellar medium (ISM) the amount of gas and dust affects the physical mecha-nisms of the ISM and can affect the evolution of an entire galaxy. In particular, the amount of molec-ular gas in a cloud, cloud complex, spiral arm, or galaxy constrains the number of stars that form and the way that they form. The usual molecule for es-timating molecular gas masses has been, and still is, CO (e.g., see IAU Symp. #170, 1997, and ref-erences therein). Specifically, observations of the J = 1→0 rotational line of the isotopologue,12_C16_O (just CO for short), permit simple, but crude,

esti-mates of the mass of molecular hydrogen in an astro-nomical source. The velocity-integrated brightness, I(CO), often called theintegrated intensity, is multi-plied by a standard conversion factor, N(H2)/I(CO), to yield the molecular hydrogen column density, N(H2), which gives the H2 mass of the source af-ter integrating over the source’s projected area. The most current value of this conversion factor is about 2×1020_H2_cm−2·(K·km·s−1)−1 for the molecular

gas in the disk of our Galaxy (Dame, Hartmann, & Thaddeus 2001). Why the CO J = 1→0 line should yield an estimate of column density is far from clear. As is well known (e.g., see Evans 1980; Kutner 1984; Evans 1999), the CO J = 1 → 0 line is optically

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thick, obfuscating any simple explanation as to why it should probe molecular gas column densities.

Other tracers of molecular gas mass exist, tracers that do not possess the serious uncertainties posed by CO J = 1 → 0. While potentially simpler to use for determining column densities, the optically thin lines of the isotopologues 13_C16_{O and} 12_C18_O (just 13_{CO and C}18_{O in short form) are normally} factors of about 3 to 50 weaker than the CO J = 1 → 0 line (e.g., Kutner 1984; Langer & Penzias 1990; Nagahama et al. 1998; Maddalena et al. 1986); the12_{CO lines are better for mapping large areas of} molecular gas or for detecting weak sources, such as high-redshift galaxies (e.g., Brown & Vanden Bout 1992; Barvainis et al. 1997,1998; Alloin, Barvainis, & Guilloteau 2000; Carilli et al. 2002a). This makes the J = 1→0 line of CO, and the N(H2)/I(CO) factor, more useful or even essential in estimating the total molecular gas mass in some sources, resulting in a strong incentive for understanding the N(H2)/I(CO) factor’s behavior.

The usual attempts at accounting for why the N(H2)/I(CO) factor, or X-factor, is relatively con-stant on multi-parsec scales are variations of the ex-planation given by Dickman et al. (1986), hereafter DSS86 (e.g., Sakamoto 1996). A summary of the DSS86 explanation follows. If TR is the peak

radia-tion temperature of the CO J = 1→0 line and ∆v is the appropriately defined velocity width of this line, then I(CO) = TR∆v. If the molecular gas under

observation is virialized, then the observed velocity width is related to the mass of this gas and, there-fore, to the gas column density averaged over the solid angle subtended by the observed gas. It was then easy to show that N(H2)/I(CO) ∝ n0.5_/_T

R.

The n was the gas density averaged over the viri-alized volume of gas. DSS86 found that n had to be∼f ew×102_H2_cm−3 to give the observed value

of N(H2)/I(CO); therefore it was assumed that this volume included entire clouds. Even if the gas does not fill the beam (a point to which we will return later), we would have TR roughly proportional to

the gas kinetic temperature, TK, and we would still

have N(H2)/I(CO)∝n0.5_/_T

K. It is argued that the

quantity n0.5_/_T

K does not strongly vary on

multi-parsec scales, especially due to the weak dependence on density, resulting in a fairly stable value of X. Observational evidence does indeed seem to support a roughly constant value of the X-factor to within a factor of about 2 for the disk of our Galaxy, where X ' 2 × 1020_cm−2 · (K · km · s−1)−1 (see, e.g.,

Dame et al. 2001; Strong et al. 1988, and references therein), although the observations of Sodroski et al.

(1994) and Strong et al. (2004) suggest a higher value of X in the outer disk (a claim that is at odds with Carpenter, Snell, & Schloerb 1990). The values of the X-factor that apply to the disks of other spiral galaxies are often within factors of about 3 of that of the Galactic disk X-factor (e.g., Young & Scoville 1982; Adler et al. 1992; Gu´elin et al. 1995; Nakai & Kuno 1995; Brouillet et al. 1998; Rand, Lord, & Higdon 1999; Meier, Turner, & Hurt 2000; Meier & Turner 2001; Boselli, Lequeux, & Gavazzi 2002; Rosolowski et al. 2003). The goals of the current paper are to address the deficiencies of the DSS86 explanation of the X-factor and improving upon this explanation. Improvements are necessary because DSS86 has the following problems:

1. No treatment of radiative transfer. This is a fundamental problem with DSS86. At first glance, it might seem superfluous to treat radia-tive transfer in the optically thick case. How-ever, if we consider a clumpy medium, where the clumps can have optically thin edges and optically thin frequencies in their line profiles, then treating radiative transfer is essential for understanding the X-factor. In particular, the optically thin limit of CO J =1 →0 must also be included. Any complete treatment must in-clude the optically thin case, whether this case is observed in nature or not. This case cannot be included easily in the DSS86 explanation be-cause it includes the virial theoremwithout in-cluding radiative transfer —virialization by it-self says nothing about the optical depth of the emission.

2. Sensitivity to TK and n(H2). As discussed

in Wall (2006), I(CO) and the X-factor esti-mate the molecular hydrogen column densities to within factors of about 2 of the values ob-tained from optically thin tracers for the ma-jority of positions in the Orion clouds. In gen-eral, we know that molecular cloud kinetic tem-peratures and densities have a full range of an order of magnitude on multi-parsec scales (cf. Sanders, Scoville, & Solomon 1985; Sakamoto et al. 1994; Helfer & Blitz 1997; Plume et al. 2000). Since the X-factor supposedly varies as

n0.5_/_T

K, the temperature and density variations

caneach change X by factors of 3 to 10 (unless

n0.5_{were to vary like T}

K, but this is unlikely to

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like X ∝(n/TK)

0.3_{, would resolve this}

sensitiv-ity problem; variations of an order of magnitude in eithernor TK would allow X to vary by less

than a factor of 2.

3. Virialization of entire clouds. DSS86 require low densities (i.e., n(H2)∼f ew×102_cm−3) to

obtain the observed value of the X-factor. Given that the critical density of the CO J = 1→ 0 transition is∼3×103_H

2cm−3, the densities of

the CO-emitting structures are about an order of magnitude higher (also see the average den-sities of the filaments found by Nagahama et al. 1998).

4. Stronger dependence of peakTR on N(H2) than

of ∆v on N(H2) is not explained. DSS86 re-quire that the observed velocity width of the line depends on the gas column density. How-ever, there is evidence that it is the peak radi-ation temperature, TR, that depends on N(H2)

and that ∆v has only a weak dependence on N(H2) (see Wall 2006; Heyer, Carpenter, & Ladd 1996).

The purpose of the current paper is to propose an im-proved approach for understanding the X-factor that will resolve, or at least mitigate, the problems with DSS86. For example, the explanation proposed here includes radiative transfer in a clumpy medium and shows how the optically thick CO J = 1→0 emis-sion of a cloud can be sensitive to the optical depths of the individual clumps. As a result, this explana-tion will permit, in some circumstances, a very weak dependence on TK and n(H2). Also, even though we

will also use the virial theorem (in its simplest form), we can apply it to scales smaller than entire clouds. And the X-factor in the current proposed explana-tion will lose its dependence on the virial theorem in the optically thin case. In addition, the proposed approach will naturally explain the dependence of the peak TR on N(H2). This improved approach has

shortcomings of its own, but nevertheless may rep-resent an important step forward in understanding the X-factor.

2. A FORMULATION FOR THE X-FACTOR

2.1. Radiative Transfer in a Clumpy Cloud

The X-factor may yield a reasonable estimate of the molecular gas column density, because the in-tegrated intensity of the CO J = 1 → 0 line is es-sentially counting optically thick clumps in the gas in the beam (e.g., see Evans 1999). This explana-tion does not, by itself, directly relate the masses of

individual clumps to the observed integrated inten-sity, because, again, the clumps are optically thick in the CO J = 1→0 line. Applyingonly the DSS86 approach to the clumps will not work, because, as discussed in the introduction, DSS86 and the ob-served value of the X-factor together require densi-ties an order of magnitude lower than those found in the clumps of real clouds. The DSS86 derivation of the X-factor depends on the beam-averaged col-umn density, N, and the observed velocity width, ∆v. We need a treatment of the problem in which the beam-averaged quantities, N and ∆v, are can-celled out in favor of the corresponding quantities for an individual clump, i.e.,Nc and ∆vc. And we need a treatment of the radiative transfer in a clumpy medium. Martin et al. (1984) (hereafter MSH84) developed a method for describing radiative trans-fer through a clumpy medium in a highly simpli-fied case: they assumed that each clump was ho-mogeneous and in LTE. For additional simplicity, they also assumed that the clumps were identical, although they pointed out that their method could be easily generalized to clumps with a spectrum of properties (see the Appendix of MSH84). The as-sumption of LTE was necessary because the implicit assumption is that the excitation temperatureof the transitionis constant throughout each clump, which is easily attained if the density is high enough for LTE. This assumption is particularly appropriate for the CO J = 1→ 0 line: because of its high optical depth (i.e., τ ∼ f ew) and low critical density (i.e.,

ncrit '3×10

3_cm−3), this line is largely thermalized

(i.e., close to LTE). Hence the method of MSH84 is appropriate here.

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pressed as the product of the number of clumps per clump velocity width on a sightline and the effective optical depth of an individual clump. If N is the beam-averaged gas column density and Nc is the gas column density averaged over the projected area of a single clump, then (N/Nc)(∆vc/∆v) is the num-ber of clumps per sightline averaged over the beam per clump velocity width at the line central velocity. (Note that, below, Nc is actually the column density on the central sightline through the clump, but this change in definition accords with the definition of the clump average optical depth.) Ifτ0is the optical depth on a sightline through the center of a single clump, then, following MSH84, A(τ0) is the clump effective optical depth. (Note that MSH84 called A(τ0) the effective optically thick area of the clump. Even though their term is more accurate, the sim-pler “clump effective optical depth” is adopted here.) Consequently,

τef(vz) = N Nc

∆vc

∆v A(τ0) exp

− vz

2

2 ∆v2

, (1)

wherevz is the velocity component along the sight-line and where a Gaussian sight-line profile has been as-sumed. Ifτ(x, y) is the clump optical depth on the sightline at position (x,y) with respect to a sightline through the clump center, then the clump effective optical depth is given by

A(τ0) = √ 1

2π∆vc aef f

Z

dv

Z

dx

Z

dy·

·

1−exp

−τ(x, y) exp

− v

2

2 ∆v2 c

, (2)

whereaef f is clump’s effective projected area defined

in terms of its optical depth:

aef f ≡

1

τ0 Z

dx

Z

dy τ(x, y) . (3)

Theτ0 is simply τ(x= 0, y= 0), the optical depth through the clump’s center and at the center of the clump’s velocity profile. The integrals in both (2) and (3) are over the projected area of the clump and over the clump’s velocity profile. Note that, in general, the expression for τef(vz) would have inte-grals over both the line spectral distribution (over

vz) and over the velocity distribution of each indi-vidual clump (overv). But, as done in MSH84, we assume that ∆v is significantly larger than ∆vc and all clumps are assumed to be identical. This then leads to expressions (1) and (2). Numerically inte-grating equation (2) gives Figure 1, which shows the

Fig. 1. The effective optical depth of a clump, A(τ0), after averaging over its projected area is plotted against the optical depth through the clump center, τ0. The solid curve shows A(τ0) versusτ0 for a cylindrical clump viewed perpendicularly to the symmetry axis. The opti-cal depth profile across the cylinder, from the central axis towards the edges, is Gaussian. The dashed curve shows the corresponding curve for a spherical clump. The op-tical depth profile from the sphere’s center towards the edges is also Gaussian. The Gaussian spherical clump case was also treated and plotted in MSH84.

variation of A(τ0) as a function of τ0 for two types of clumps: cylindrical (seen orthogonally to the axis of symmetry) and spherical.

The observed line radiation temperature, TR, is

then related toτef by the usual expression

TR(ν) =Jν(TK) [1−exp(−τef)] . (4)

TK is the gas kinetic temperature. As stated

ear-lier, LTE is assumed for the emission of the spectral line at frequency ν. The Jν(TK) is correction for

the failure of the Rayleigh-Jeans approximation and for the cosmic microwave background emission. Of course whenτef 1, we have the simplified form of Eq. (4):

TR(ν) =Jν(TK)τef . (5)

This is often called the “optically thin limit” for the equation of radiative transfer. An important point here is that the effective optical depth is in the op-tically thin limit even though the individual clumps can still be quite optically thick. And this, of course, will provide a partial explanation for the X-factor. Substituting Eq. (1) into Eq. (5) yields

TR(vz) =Jν(TK)

N Nc

∆vc

∆v A(τ0) exp

− vz

2

2 ∆v2

.

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As mentioned previously, the quantity (N/Nc)(∆vc/∆v) is the number of clumps per sightline averaged over the beam within a clump velocity width at the line central velocity. This quantity can be much less than unity, thereby permitting τef 1 even for A(τ0) 1. (When (N/Nc)(∆vc/∆v)<1, it is similar to the geometric area filling factor within a narrow velocity interval, although it is not necessarily equivalent.) Since A(τ0) is roughly equivalent to [1−exp(−τ)] for a single clump, the meaning of expression (6) is clear: it is the specific intensity of a single clump at veloc-ityvz—∼ Jν(TK) [1−exp(−τ)] —multiplied by the

number of clumps at that velocity within a clump ve-locity width— (N/Nc)(∆vc/∆v) exp[−vz2/(2 ∆v2)]. Simply multiplying the intensity of a single clump by the number of clumps gives the observed inten-sity if the clumps are radiatively de-coupled, and this is ensured if τef 1. As τef increases and becomes optically thick, the different clumps within each velocity interval start absorbing each other’s emission and the radiation temperature approaches the source functionJν(TK) asymptotically.

We can now better understand the behavior of the curves in Fig. 1. A(τ0) represents the level of emission from a single clump averaged over the clump’s projected area. Whenτ0 1, A(τ0) ' τ0 because all lines of sight through the clump and the line’s profile at all the clump’s internal velocities are optically thin. Asτ0increases past unity, A(τ0) con-tinues growing because of those lines of sight and those velocities at which the emission is still optically thin. Fig. 1 shows two curves: one for a cylindrical clump of gas (i.e., a filament) and one for a spherical clump. The cylinder is viewed side-on (i.e. with its symmetry axis perpendicular to the sightline) and has length h. If the symmetry axis is the x-axis, then a Gaussian variation of the optical depth with

y was adopted:

τ(x, y) =τ0exp

−πh2 y

2

aef f

2

. (7)

The spherical clump also has Gaussian spatial varia-tion with optical depth, but with radial distance,p, from the central sightline through the clump:

τ(x, y) =τ0exp

−π p

2

aef f

, (8)

where p = px2₊_y2_{. This case was also treated} by MSH84, and it is included here for comparison. (Note that the aef f used here corresponds to the r2

o of MSH84.) The effective optical depth of the

spherical clump grows faster withτ0forτ0 >_∼1 than that of the cylindrical clump because the former’s optically thick area is growing simultaneously in two dimensions, whereas the latter’s grows only in one. While A(τ0) can grow without bound in these ideal-ized cases, TR cannot. Eventually, A(τ0) will grow

large enough that τef 1 is no longer valid and TR asymptotically approaches Jν(TK). The

grow-ingτ0causes this to happen because the clumps start crowding each other spatially and in velocity, due to their increasing optically thick areas and their in-creasingly saturated line profiles.

The curves of Fig. 1 demonstrate that we can represent them as power-laws in τ0 for τ0 ≤ 1 or

τ0≥3:

A(τ0)'kAτ

0 . (9)

The values ofkA and obviously depend on the specificτ(x, y) —the opacity structure of the clump, except in the optically thin case. Whenτ0 <1, we have kA = 1 and = 1, regardless of the specific variation of τ(x, y). Except for the optically thin case, a lower value of, i.e., closer to zero, indicates a clump with a better defined outer edge, like a hard sphere. Conversely, a higher value of , i.e., closer to unity, indicates a clump with a more tenuous, or fluffier, outer region. Accordingly, will be called the “fluffiness” of the clump.

2.2. Relating Clump Velocity Width with Column Density

DSS86 required virialization in order to relate the line velocity width to the gas column density. That is also required here, but it will be combined with the radiative transfer in a clumpy cloud discussed in the previous subsection. The virial theorem in its simplest form neglects the effects of surface pressure and magnetic fields. Assuming a spherical clump of uniform density gives

∆vc = kvN0.5c L0.5c , (10)

and

kv ≡ π

15G µmH2

0.5

. (11)

Numerically incgs units, this is

kv = 2.47×10−16 ,

where µ is the correction for the mass of helium in the ISM and µ = 1.3 was used. The Nc and Lc are the column density and path length through the clump central sightline, respectively. The mH2 is the

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2.3. Relating Clump Optical Depth with Column Density

The clump optical depth on the sightline through the clump’s center,τ0, can be written in terms of the column density of CO in levelJ,NJ:

NJ =

8π AJ,J₋1λ

3

J,J−1 ×

×

"

exp TJ,J−1

TK

!

−1 #−1

τ0

√

2π∆vc .(12)

This comes from Eq. (A9) of Wall (2006) after ap-plying the Boltzmann factor to changeNJ−1toNJ.

The velocity integral was replaced by τ0

√

2π∆vc, where τ0 is the optical depth at the center of the clump’s velocity profile and on the sightline through the clump’s center. TJ,J−1 is the energy of the

J → J −1 transition in units of temperature: i.e.,

TJ,J−1 =h νJ,J−1/k with νJ,J−1 as the frequency of

the transition. AJ,J−1andλJ,J−1are the spontaneous

transition rate and the wavelength of the transition, respectively. LTE is assumed, so TKapplies in place

of TX(J →J−1). We can determine the total

col-umn density of CO,N(CO), by substituting Eq. (12) into Eq. (A22) of Wall (2006) and rearranging:

τ0 =

3A10λ310 8√2π32∆vcQ(T

K) ×

×

"

1−exp −T10

TK

!#

NcX(CO) . (13)

whereQ(TK) is the partition function of CO and J

was set to 1. TheN(CO) was replaced by NcX(CO), whereX(CO) is the abundance of CO relative to H2. The following values are used (see Wall 2006, and references therein): A10 = 7.19×10−8s−1, T₁₀ =

5.54 K, λ10 = 0.2601 cm, and X(CO) = 8×10−5.

Accordingly,

τ0= 1.21×10

−14

√

2π∆vcQ(TK)

"

1−exp −5.54

TK

!# Nc .

(14) The above expression can be represented more sim-ply as a power-law in TK:

τ0=

kτ

√

2π∆vcNcT

−γ

K . (15)

The exact values of kτ and γ depend on the tem-perature range and can be computed by numeri-cally comparing expressions (15) and (14). In the

high-temperature limit, however, an analytical so-lution is possible. This limit means that TK T10

and Q(TK)→ 2TK/T10 and [1−exp(−T10/TK)]→ T10/TK. This results in kτ = 1.85×10−13 in cgs units and γ = 2. But we will be interested in the temperature range TK = 10 to 20 K. The necessary

numerical comparison gives us

kτ = 7.30×10−14 (cgs units) and

γ = 1.75,

for that range. This approximation is accurate to within 1–2% in the above specified range. Equa-tion (15) simplifies further by using expression (10) for ∆vc andnc = Nc/Lc:

τ0=

kτ

kv

√

2πn

0.5

c T−Kγ . (16)

This interesting result suggests that the optical depth for this simplified case (i.e., the velocity pro-file of the optical depth is a simple Gaussian) of a virialized clump does not explicitly depend on the sightline pathlength nor the velocity width, but on their ratio. This is related to the Sobolev approx-imation (e.g., see Shu 1991) in which the optical depth is dependent on the velocity gradient within a given region and not explicitly on the region’s size. The pathlength-to-velocity-width ratio (Lc/∆vc) in a virialized clump is determined by the average den-sity, the spatial variation of the denden-sity, and the geometry. Therefore, the optical depth depends on those things and the gas temperature, but with no dependence on the clump size or velocity width (at least for this simplified case).

2.4. The X-Factor

Understanding how to combine the results of the previous subsections to derive an expression for the X-factor requires examining the observational data that inspired the current paper in the first place. Figure 2 shows the Orion data discussed in Wall (2006): the peak radiation temperature of the 12_{CO J = 1}_→_{0 line (i.e., T}

R) for various positions in

the Orion clouds normalized to the source function at each position (i.e., Jν(TK), the source function

in temperature units) versus the gas column density (i.e., N(H2)) as determined from 13_{CO J = 1} _→ _0. The plots demonstrate a clear correlation between TR/Jν(TK) and N(H2). The Spearman rank-order

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J = 1 → 0 lines of 12_{CO and} 13_{CO, because the}

Jν(TK) is determined from the dust temperature

(see Wall 2006, for details). This suggests that the dust temperature really is a reliable measure of the kinetic temperature of the molecular gas, at least for the Orion clouds on the scales of parsecs (see Wall 2006,a,b, for more discussion of this). One way of ex-plaining the correlation visible in Fig. 2 is that the area filling factor of the clump in each clump velocity interval is less than unity. A rising beam-averaged column density,N, means the filling factor is rising as more and more clumps fill the beam in each ve-locity interval. Obviously, the goal here is to be very specific about the relationship between TR/Jν(TK)

and N(H2). There is sufficient scatter and uncer-tainty in the data that it is not possible to rule out a priori a number of such relationships.

Nevertheless, from simple radiative transfer the-ory, we know that the specific intensity of a source normalized to its source function, usually written

Iν/Sν, will vary like 1 −exp(−τ) when plotted against the optical depth through the source, τ. Given that the column density, N, is proportional to τ for constant kinetic temperature and density, the data in Fig. 2 mimic a curve with the form 1−exp(−aN) (see the plotted curves), where the

aN probably represents some kind of optical depth. The majority of the data points are on the roughly linearly rising portion of the curve. This repre-sents the optically thin region of the curve, but the 12_{CO J = 1} _→ ₀ _{is known to be optically thick}

from comparisons with the optically thin isotopo-logue13_{CO. Therefore,}_{a clue to understanding the} X-factor is realizing that COJ = 1→0emission be-haves like it is optically thin, despite being optically thick. This apparent contradiction is resolved when we consider the effective optical depth as described previously. While the individual clumps are them-selves optically thick in CO J = 1→0, the cloud is optically thin “to the clumps.” In other words, the emission from every clump in the telescope’s beam through the cloud reaches the observer. An anal-ogy would be observing the HI21 cm line from an atomic cloud. In this case, the cloud is optically thin “to the atoms” in the sense that the emission from every atom in the telescope’s beam through the cloud reaches the observer. And since every hydro-gen atom is nearly identical in its 21 cm line emission properties, the conversion from I(HI) to N(HI) is physically straightforward and undisputed. For con-verting from I(CO) to N(H2), assuming absolutely identical clumps would give a constant value of the X-factor, relatable to the clumps’ properties. But

Fig. 2. Plots of the CO J = 1→0 line radiation tem-perature, TR, normalized to its source function,Jν(TK),

versus the 13

CO J = 1 →0 derived H2 column density. Both plots are reproduced from Wall (2006). The curves are of the form y = 1−exp(−a x−b), where, in the ideal case, b = 0. The upper plot is for the LVG, one-component models of Wall (2006) and the lower plot is for the LVG, two-component, two-subsample models of that paper. (The source function for the two-component models is the effective source function as defined in Wall 2006). Given that x is in units of 1020

H2cm−2_, _a ₌

(9.5±0.4)×10−3_and_b_{= (2}_.₄_±₅_.₅₎_×₁₀−3_{for the upper}

plot anda= (6.4±0.2)×10−3_and_b_{= (7}_.₂_±₀_.₈₎_×₁₀−2

for the lower plot.

we need not restrict the clump properties so severely to explain the X-factor. All we need is to have the clumps similaron averagefrom one beam to the next for the X-factor to stay relatively constant.

We now need to quantify this picture, so that we might better understand it and its limitations. As Fig. 2 clearly shows, the TR/Jν(TK) ∝ N for

N <_∼1 to 2×1022_H2_cm−2. This is in the τ

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A(τ0)— and the observed line width, ∆v, do not vary strongly withN. In fact, the scatter visible in the plots of Fig. 2 is probably due to variations in all four of these quantities. Integrating Eq. (6) over velocity,vz, gives

I(CO) = √2πTR(0) ∆v,

= √2πJν(TK)τef(0) ∆v . (17)

TR(0) is the radiation temperature of the CO J =

1→0 line at vz = 0 and is also the peak radiation temperature of this line. Similarly, τef(0) is the ef-fective optical depth atvz= 0. The X-factor is then given by

Xf = h√

2πJν(TK)τef(0) ∆v N−

1i−1 _. ₍₁₈₎

If we now substitute Eq. (1) evaluated atvz= 0 into the above, then

Xf = h√

2πJν(TK) A(τ0) ∆vcN

−1

c i−1

.(19)

The important thing to notice here is that the di-rectly observed quantities,N and ∆v, have been re-placed by the corresponding clump properties, Nc and ∆vc. In fact, all the parameters in expres-sion (19) are clump parameters, as desired. It is convenient to define

CT ≡ TK Jν(TK)

. (20)

We now use the approximationJν(TK)'TK−3.4 K

for TK>∼10 K and the frequency of the

12_{CO J = 1}_→

0 line,ν= 115.271 GHz. This is good to within 0.4% of TK (and within 0.6% ofJν(TK)). Consequently,

CT =

TK

TK−3.4 K

, (21)

which approaches unity as TK grows large. Now we

substitute the results of the previous subsections into equation (19): equation (9) for A(τ0), (16) for τ0, (10) for ∆vc, and TK/CT for Jν(TK). We also use

nc= Nc/Lc. These substitutions yield

Xf = (2π)

1 2(−1)_C

Tk−A1k−τk− 1

v TγK−1n 1 2(1−)

c .

(22) The above formulation for Xf obviously accom-plishes the goal of insensitivity to the parameters TK and nc that we have sought for Xf. The

fluffi-ness parameter, , is in the range 0 to 1; any value in that range that is greater than 0 will confer a greaterinsensitivity than occurs for the DSS86 ex-planation. A particularly interesting example is that

value offor whichγ−1 = 0. In the high temper-ature limit, CT → 1 and γ → 2 and, if = 0.5, thenXf has no dependence on temperature. (Notice that the density dependence is also very weak in this case: Xf ∝n0.25c .) Given that the CO J = 1 → 0 line is optically thick, this is counterintuitive; rais-ing the temperature by some factor should simply increase I(CO) by the same factor (in the high-TK

limit), thereby decreasingXf by that factor. That is not the case here. Here we are dealing with a clumpy medium where the optical depth varies across the projected area of each clump. There will always be some sightlines through a clump that will still be op-tically thin. There will also be some velocities in the clump’s spectral line profile where the line emission is still optically thin. As TK increases, τ0 goes like

T−2

K , so that A(τ0) goes like T

−1

K (see Eqs (15) and

(9). ButJν(TK) goes like T

1

K(in this high-TKlimit),

meaning that the observed TR stays constant. The

effect of the increasing kinetic temperature of the gas is cancelled by the decreasing effective optical depth of the clumps. Another way of saying this is that the effect of the rising temperature is cancelled by the shrinking effective optically thick areas of the clumps; the filling factors of the clumps decline as the temperature rises. (Note that this special case also occurs for lower temperatures. For TK= 10 to 20 K,

for example,CT ∝T−K0.34andγ= 1.75. The value of

for whichXf is independent of temperature would be 0.77.) Therefore,despite the optical thickness of the spectral line in the emitting clumps, changing the optical depths of the individual clumps will still have an appreciable effect on the line strength. And this will reduce the dependence of the X-factor on the temperature and the density of the gas within the clumps.

3. CONCLUSIONS

What this treatment means in the astronomical context will be examined in more detail in a future paper (i.e., Wall 2006c). For now, it can be stated that the goals set out in the introduction have been largely accomplished:

(9)

2. Reduced Sensitivity to TK and n(H2). As the

example in the previous section illustrates, the sensitivity to clump density and temperature can be greatly reduced or even eliminated (for the case of TK only). For the case of a

spher-ical clump with a Gaussian spatial variation of the optical depth, this dependence is roughly,

Xf ∝(nc/TK)

0.3_.

3. Virialization of entire clouds is not necessary. By using densities more representative of indi-vidual clumps, i.e., nc ∼ f ew×103cm−3, the treatment given here will yield X-factor values easily within factors of 2 of those observed in our Galaxy.

4. Stronger dependence of peakTR on N(H2) than

of ∆v on N(H2) is now explained. Given that the X-factor in this treatment depends on the velocity width of the individual clumps and not on the observed velocity width, ∆v, of all the gas in the beam, the lack of dependence on ∆v is now explained.

However, despite these successes, problems still remain:

1. Sensitivity to. The previous sensitivity to the physical parameters of density and temperature has now been replaced by the strong dependence on yet another physical parameter: . Why this parameter should remain roughly constant is uncertain. Nevertheless, it’s variation maybe one reason why the X-factor can be very differ-ent in certain special regions such as the Galac-tic Centre region (e.g., Sodroski et al. 1995; Dahmen et al. 1998).

2. Clump optical depths too high.Given reasonable temperatures and densities (i.e. TK ∼10–20 K

andnc ∼f ew×103cm−3) for the clumps, the central sightline optical depths in the12_{CO J =} 1 → 0 line are an order of magnitude higher than expected from observations of the optically thin 13_{CO J = 1} _→ _{0 line. However, the} X-factor is only reasonably constant on the scales of many parsecs. On such scales, the central sightline optical, τ0, is not really the relevant quantity; the average over the clump’s projected area, A(τ0), is the relevant quantity and is about an order of magnitude smaller.

A more detailed discussion of the problems will be given in Wall (2006c). Overall, the treatment holds much promise, given that it includes basic ra-diative transfer.

This work was supported by CONACyT grants #211290-5-0008PE and #202-PY.44676 to W. F. W. at INAOE. I am very grateful to W. T. Reach for his comments and support. I thank T. A. .D. Paglione, G. MacLeod, E. Vazquez Semadeni, and others for stimulating and useful discussions. The author is grateful to R. Maddalena and T. Dame, who sup-plied the map of the peak 12_{CO J = 1} _→ _{0 line} strengths and provided important calibration infor-mation.

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